Optimization Concepts for Resource Allocation in Cellular Systems

Resource Allocation inCellular SystemsEmil BjrnsonPhD in Telecommunications

Signal Processing LabKTH Royal Institute of TechnologySuplec, 2012-02-02Optimization Concepts forKTH in StockholmKTH was founded in 1827 and is the largest of Swedens technical universities.

Since 1917, activities have been housed in central Stockholm, in beautiful buildings which today have the status of historical monuments.

KTH is located on five campuses.

22012-02-02Emil Bjrnson, KTH Royal Institute of Technology3A top European grant-earning universityEuropes most successful university in terms of earning European Research Council Advanced Grant funding for investigator-driven frontier research

5 research projects awarded in 2008:Open silicon-based research platform for emerging devices Astrophysical Dynamos Atomic-Level Physics of Advanced Materials Agile MIMO Systems for Communications, Biomedicine, and Defense Approximation of NP-hard optimization problems

2012-02-02Emil Bjrnson, KTH Royal Institute of TechnologyEmil Bjrnson1983: Born in Malm, Sweden

2007: Master of Science inEngineering Mathematics,Lund University, Sweden

2011-11-17: Defended doctoralthesis in telecommunications,KTH, Sweden

Multiantenna Cellular CommunicationsChannel Estimation, Feedback, and Resource Allocation

Three Building Blocks of Physical LayerMathematical Analysis and Optimization2012-02-02Emil Bjrnson, KTH Royal Institute of Technology4

BackgroundCellular CommunicationsMany transmitting multi-antenna base stationsMany receiving single-antenna usersDownlink TransmissionMultiple transmit antennas exploit spatial dimensionMultiuser transmission Pro: Higher performance, Con: Co-user interference2012-02-02Emil Bjrnson, KTH Royal Institute of Technology5

Background (2): Multiple CellsUncoordinated Cells: Simple processing Simple infrastructure Uncontrolled interference Or fractional frequency reuse

Coordinated Cells: Controlled interference Backhaul signaling Computationally complex Tight synchronization2012-02-02Emil Bjrnson, KTH Royal Institute of Technology6

Background (3): Multiple CellsDynamic User-Centric Coordination Clusters Inner Circle (Strong Channels):Consider Transmitting to UsersOuter Circle (Non-negligible Channels):Avoid Interference to UsersCan model any level of coordination

2012-02-02Emil Bjrnson, KTH Royal Institute of Technology7

Background (4)Resource AllocationSelect users for transmissionDesign beamforming directionsAllocate transmit power

Optimize Resource AllocationMaximize system performanceSatisfy system constraints (power, interference, fairness)Any level of coordinated multipoint transmissionRobustness to uncertain channel information

No mathematical detailsFocus on performance optimization conceptsAssumption: Linear transmit/receive processing2012-02-02Emil Bjrnson, KTH Royal Institute of Technology8OutlineWhat is Performance?Different user performance measuresSystem Performance vs. user fairness

Multi-user Performance RegionHow to interpret?How to choose operating point?

Performance OptimizationGeometrical interpretation of common formulationsRight problem formulation = Easy to solve

Low-complexity StrategiesExploit structure from optimal solution2012-02-02Emil Bjrnson, KTH Royal Institute of Technology92012-02-02Emil Bjrnson, KTH Royal Institute of Technology10What is Performance?What is Performance?Service QualityExperienced by users (per-user level)Can also be measured at system-level

Performance Based onAverage data rateLatencyCoverageBattery lifeEtc.

Simplified Performance MeasuresNecessary for optimization2012-02-02Emil Bjrnson, KTH Royal Institute of Technology11Single-user Performance MeasuresMean Square Error (MSE)Difference: transmitted and received signalEasy to analyzeFar from user perspective?

Bit/Symbol Error Rate (BER/SER)Probability of error (for given data rate)Intuitive interpretationComplicated & ignores channel coding

Data RateBits per channel useMutual information: perfect and long codingStill closest to reality?

2012-02-02Emil Bjrnson, KTH Royal Institute of Technology12All improveswith SNR:

Signal PowerNoise Power

Optimize SNR instead!12Multi-user PerformanceUser Performance MeasuresSame measures but one per user

Performance LimitationsPower AllocationCo-user interference: SINR=

Why Not Increase Power?Power = Money & Environmental ImpactReduce noise Interference limited

User FairnessNew dimension of difficultyHeterogeneous user conditionsDepends on performance measure

2012-02-02Emil Bjrnson, KTH Royal Institute of Technology13Signal PowerInterference + Noise Power

2012-02-02Emil Bjrnson, KTH Royal Institute of Technology14Multi-user Performance RegionMulti-user Performance Region2012-02-02Emil Bjrnson, KTH Royal Institute of Technology15

Performance User 1Performance User 2PerformanceRegionCare aboutuser 2Care aboutuser 1BalancebetweenusersAchievable Performance Region 2 users - Under power budgetPart of interest:Upper boundary

Multi-user Performance Region (3)Can it have any shape?

No! Can prove that:Compact setSimply connected (No holes)Nice upper boundary2012-02-02Emil Bjrnson, KTH Royal Institute of Technology16Normal setUpper corner in region, everything inside regionMulti-user Performance Region (3)Possible Shapes of RegionConvex, concave, or neitherIn general: Non-convexIn any case: Region is unknown2012-02-02Emil Bjrnson, KTH Royal Institute of Technology17

Convex

Concave

Non-convexNon-concave

Multi-user Performance Region (3)Some Operating Points Game Theory Names2012-02-02Emil Bjrnson, KTH Royal Institute of Technology18

Performance User 1Performance User 2PerformanceRegionUtilitarian point(Max sum performance)

Egalitarian point(Max fairness)

Single user point

Single user point

Which pointto choose?

Optimize:Sum Performance?Fairness?2012-02-02Emil Bjrnson, KTH Royal Institute of Technology19Performance OptimizationSystem Performance versus FairnessAlways Sacrifice EitherSum PerformanceUser FairnessOr both: optimize something in betweenTwo Standard Optimization StrategiesMaximize weighted sum performance: maximize w1R1 + w2R2 + (w1 + w2+ = 1)

Maximize performance with fairness-profile: maximize Rsum subject to R1=a1Rsum, R2=a2Rsum, (a1 + a2+ = 1)

Non-Convex Optimization ProblemsGenerally hard to solve numerically

2012-02-02Emil Bjrnson, KTH Royal Institute of Technology20R1,R2,RsumStarts fromPerformanceStarts fromFairnessThe Easy ProblemGiven Point (R1,R2,)Find transmit strategy that attains this pointMinimize power usageConvex ProblemSecond-order cone or semi-definite programGlobal solution in polynomial time use CVX, Yalmip

M. Bengtsson, B. Ottersten, Optimal Downlink Beamforming Using Semidefinite Optimization, Proc. Allerton, 1999.A. Wiesel, Y. Eldar, and S. Shamai, Linear precoding via conic optimization for fixed MIMO receivers, IEEE Trans. Signal Processing, 2006.W. Yu and T. Lan, Transmitter optimization for the multi-antenna downlink with per-antenna power constraints, IEEE Trans. Signal Process., 2007.E. Bjrnson, G. Zheng, M. Bengtsson, B. Ottersten, Robust Monotonic Optimization Framework for Multicell MISO Systems, IEEE Transactions on Signal Processing, To appear.

2012-02-02Emil Bjrnson, KTH Royal Institute of Technology21Single-cell(total power)

Single-cell(per ant. power)

Multi-cell(general power, robustness)

Exploiting the Easy ProblemEasy: Achieve a Given PointHard: Find a Good Point

Shape of Performance RegionFar from obvious one dimension per user

2012-02-02Emil Bjrnson, KTH Royal Institute of Technology22Rate: user 3Rate: user 1Rate: user 2Interference Channel

3 transmittersw. 4 antennas

3 usersMain part of resource allocation

Geometric Optimization InterpretationsMaximize Performance with Fairness Profile: maximize Rsum subject to R1=a1Rsum, R2=a2Rsum, (a1 + a2+ = 1)Geometric InterpretationSearch on line in direction (a1,a2,) from origin2012-02-02Emil Bjrnson, KTH Royal Institute of Technology23(a1,a2,)Rsum =(a1Rsum,a2Rsum,)RsumGeometric Optimization Interpretations (2)2012-02-02Emil Bjrnson, KTH Royal Institute of Technology24

Simple line-search algorithm: BisectionNon-convex Iterative convex (Quasi-convex)Find start intervalSolve the easy problem at midpointIf feasible: Remove lower halfElse: Remove upper halfIterate

Subproblem: Convex optimizationLine-search: Linear convergenceOne dimension (independ. #users)

Geometric Optimization Interpretations (3)Maximize weighted sum performance: maximize w1R1 + w2R2 + (w1 + w2+ = 1)Geometric interpretationSearch on line w1R1 + w2R2 = max-value2012-02-02Emil Bjrnson, KTH Royal Institute of Technology25Max-value is unknown!Distance from origin unknownLine hyperplane (dim: #user 1)Harder than fairness-profile problem!Iterative search algorithm?R1,R2,Geometric Optimization Interpretations (4)Systematic Search AlgorithmConcentrate on important parts of performance regionImprove lower/upper bounds on optimum:

Continue until

Efficiently Solvable SubproblemsBased on Fairness-profile problem

2012-02-02Emil Bjrnson, KTH Royal Institute of Technology26

Branch-Reduce-Bound (BRB) AlgorithmCover region with a boxDivide the box into two sub-boxesRemove parts with no solutions in Search for solutions to improve bounds(Based on Fairness-profile problem)Continue with sub-box with largest value2012-02-02Emil Bjrnson, KTH Royal Institute of Technology27

Geometric Optimization Interpretations (5)

2012-02-02Emil Bjrnson, KTH Royal Institute of Technology28PropertiesGlobal ConvergenceAccuracy >0 in finitely many iterationsExponential complexity only in #usersPolynomial complexity in other parameters (#antennas/constraints)Geometric Optimization Interpretations (6)

Geometric Optimization: ConclusionsFairness-Profile Approach: EasyQuasi-Convex: Polynomial complexityReason: Only one search dimensionWeighted Sum Performance: DifficultNP-hard: Exponential complexity (in #users)Reason: Optimizes both performance and fairness

Every Weighted Sum = Some Fairness-ProfileEasier to solve when posed as fairness-profile problemParameter relationship non-obvious

2012-02-02Emil Bjrnson, KTH Royal Institute of Technology29Geometric Optimization: ReferencesLine-Search Algorithm for Fairness-ProfilesM. Mohseni, R. Zhang, and J. Cioffi, Optimized transmission for fading multiple-access and broadcast channels with multiple antennas, IEEE J. Sel. Areas Commun., vol. 24, no. 8, pp. 16271639, 2006.J. Lee and N. Jindal, Symmetric capacity of MIMO downlink channels, in Proc. IEEE ISIT06, 2006, pp. 10311035.E. Bjrnson, M. Bengtsson, and B. Ottersten, Pareto Characterization of the Multicell MIMO Performance Region With Simple Receivers, IEEE Trans. on Signal Processing, Submitted, 2011.BRB AlgorithmUseful for more than weighted sum performanceE.g. arithmetic, geometric, or harmonic mean performanceH. Tuy, F. Al-Khayyal, and P. Thach, Monotonic optimization: Branch and cut methods, Essays and Surveys in Global Optimization, Springer, 2005.E. Bjrnson, G. Zheng, M. Bengtsson, B. Ottersten, Robust Monotonic Optimization Framework for Multicell MISO Systems, IEEE Transactions on Signal Processing, To appear.2012-02-02Emil Bjrnson, KTH Royal Institute of Technology302012-02-02Emil Bjrnson, KTH Royal Institute of Technology31Low-complexity StrategiesLow-complexity StrategiesHardware LimitationsPolynomial complexity: Only slowly-varying channelsExponential complexity: Only suitable for benchmarking

Heuristic Resource AllocationFind reasonable strategy with little effortExploit available insight the optimal structure

Parametrization of Upper BoundarySelect parameters in [0,1]Get an strategy explicitlyCan achieve any point on upper boundaryOnly necessary condition2012-02-02Emil Bjrnson, KTH Royal Institute of Technology32

Low-complexity Strategies (2)Method 1: Interference-temperature ControlTransmitters x (Receivers 1) parametersX. Shang, B. Chen, and H. V. Poor, Multi-user MISO interference channels with single-user detection: Optimality of beamforming and the achievable rate region, IEEE Trans. Inf. Theory, 2011.R. Mochaourab, E. Jorswieck, Optimal Beamforming in Interference Networks with Perfect Local Channel Information, IEEE Trans. Signal Processing, 2011.Method 2: Exploit Solution Structure of Easy ProblemExplicit strategy given by optimal Lagrange multipliersAlways same structure, but different parametersTake Lagrange multipliers as our parameters!Transmitters + Receivers 1 parametersE. Bjrnson, M. Bengtsson, and B. Ottersten, Pareto Characterization of the Multicell MIMO Performance Region With Simple Receivers, IEEE Trans. Signal Processing, Submitted, 2011.

2012-02-02Emil Bjrnson, KTH Royal Institute of Technology33Low-complexity Strategies (3)2012-02-02Emil Bjrnson, KTH Royal Institute of Technology34Number of ParametersLarge difference for large problems

Number of Transmitters/ReceiversLow-complexity Strategies (4)2012-02-02Emil Bjrnson, KTH Royal Institute of Technology352012-02-02Emil Bjrnson, KTH Royal Institute of Technology36ExampleExample Multicell Scenario2012-02-02Emil Bjrnson, KTH Royal Institute of Technology37Maximize Weighted Sum RateTwo base stations: 20 dBm output powerFull, Partial, or No coordination

BRB algorithmHeuristic Parameters (=1) SummaryEasy to Measure Single-user PerformanceMulti-user Performance MeasuresSum performance vs. user fairnessPerformance RegionAll combinations of user performanceUpper boundary: All efficient outcomesExplicit Parametrization: Low-complexity strategiesTwo Standard Optimization StrategiesMaximize weighted sum performanceDifficult to solve (optimally heuristic approx. exists)Maximize performance with fairness profileEasy to solve (with line-search algorithm)2012-02-02Emil Bjrnson, KTH Royal Institute of Technology382012-02-0239Emil Bjrnson, KTH Royal Institute of TechnologyThank You for Listening!

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